ILO was not failing back to lower version of TLS if TLS 1.2 was selected. After unselecting TLS 1.2 from Internet Explorer 11, I was able to connect to ILO interface. This is issue was resolved with later version ILO firmware. So, after patching the server with latest ILO firmware, I was able to connect to ILO3 interface using Internet Explorer 11 with TLS 1.2 selected.
Unable to open ILO3 with TLS 1.2
In this case I was unable to connect to ILO3 on HP DL 380 G7 with Internet Explorer 11 from Windows 8.1 client workstation. ILO Firmware version was 1.20. Starting from Windows 8.1 and Internet Explorer 11 all TLS protocols are enabled and supported by default: